UW-PIMS Mathematics Colloquium: Bianca Viray

Date: 01/13/2014

Time: 14:30

Lecturer(s):

Bianca Viray, Brown University

Location:

University of Washington

Topic:

The local to global principle for rational points

Description:

Let X be a connected smooth projective variety over Q. If X has a
Q point, then X must have local points, i.e. points over the reals and
over the p-adic completions Q_p. However, local solubility is often not
sufficient. Manin showed that quadratic reciprocity together with higher
reciprocity laws can obstruct the existence of a Q point (a global
point) even when there exist local points. We will give an overview of
this obstruction (in the case of quadratic reciprocity) and then show
that for certain surfaces, this reciprocity obstruction can be viewed in
a geometric manner. More precisely, we will show that for degree 4 del
Pezzo surfaces, Manin's obstruction to the existence of a rational point
is equivalent to the surface being fibered into genus 1 curves, each of
which fail to be locally solvable. This talk will be suitable for a
general audience.